the implementation of a lagrangian particle dispersion ... · the implementation of a lagrangian...

The implementation of a Lagrangian Particle

Dispersion Model in Large Eddy Simulations of the

Atmospheric Boundary Layer

G. J. van Dijk BSc

March 2007Master’s thesis

Supervisors:Drs. T. Heus

Dr. H. J. J. JonkerProf. Dr. Ir. H. E. A. van den Akker

Department of Multi-Scale PhysicsFaculty of Applied Sciences

Delft University of Technology

Abstract

With the growing awareness of a possibly changing climate, more and more research is per-formed on the prediction of wheater systems. In order to make these predictions better, onehas to understand, among many other things, the behaviour of small cumulus clouds. Oneof the ways to investigate shallow cumulus clouds, is by performing Large Eddy Simula-tions (LES). In an LES, the lowest few kilometers of the atmosphere are modelled.

A way to extend the possibilities of the research, is to develop a Lagrangian Particle Disper-sion Model (LPDM), that has massless particles traveling through the domain. The LPDM,developed in this thesis, runs simultaneously with the LES and uses the intermediate outputof the LES-velocities to calculate the new positions of the particles. Since these 3D-velocitiesare available only on a discrete grid, a 1st order linear interpolation scheme is used in space.With the interpolated velocities, a second order Adams-Bashfort scheme is used to calculatethe new position of the particle. Simulations have been performed with over 10 million par-ticles, where the LPDM-routine took about 30% of the total computation time. The LPDMhas been validated by comparing Eulerian statistics obtained from the LES, to Lagrangianstatistics obtained from the LPDM.

In the world of atmospheric researchers, there exists a controversy about, whether mixingof a cloud and the surrounding air can be described best by lateral mixing, or by verticalmixing. The main argument of people who defend the top-bottom mixing theory, is theconserved variable diagram, also called a Paluch diagram. In this thesis a simulation is per-formed using the LPDM where particles, that are present at the same height, at the sametime, in the same cloud are tracked back in time. It is shown that the particles originate fromall heights, with the majority originating from heights just below the observation height.This result gives good evidence for the lateral mixing concept to be the most aproperiate forshallow cumulus clouds.

3

Contents

Contents 5

1 Introduction 7

1.1 The need for parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Shallow cumulus clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Large Eddy Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Motivation for Lagrangian statistics . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 The goal of the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Structure of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 The Atmospheric Boundary Layer 11

2.1 Definitions of meteorological quantities . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Formation of shallow cumulus clouds . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 The governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Large Eddy Simulations of the ABL . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Subgrid scale model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 Remaining properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Lagrangian Particle Dispersion Model 21

3.1 Particle tracking in an LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 The interpolation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 The integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 The subgrid scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 The boundary handling scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Validation of the LPDM 29

4.1 Stability of the particle scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 The well-mixed condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Comparison of Eulerian-Lagrangian statistics . . . . . . . . . . . . . . . . . . . 32

4.3.1 The BOMEX case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.2 The CBL case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.3 The vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Computation time of the LPDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5

CONTENTS

5 Conserved variable diagrams 43

5.1 Lateral mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 Cloud-top mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 The choice of conserved variables . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4 The setup of the case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.5 Results of the case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.6 Applicability of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Conclusions and recommendations 59

6.1 The LPDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.1.1 Implementation of the LPDM . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 The LPDM in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.2.1 Validation of the LPDM scheme . . . . . . . . . . . . . . . . . . . . . . 606.2.2 Computation time of the LPDM . . . . . . . . . . . . . . . . . . . . . . 606.2.3 The timestep of the LPDM . . . . . . . . . . . . . . . . . . . . . . . . . . 606.2.4 The κ-scheme and the subgrid scheme . . . . . . . . . . . . . . . . . . . 61

6.3 Lateral or vertical mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.3.1 Conserved variable diagram . . . . . . . . . . . . . . . . . . . . . . . . 616.3.2 Setup and results from the BOMEX-case . . . . . . . . . . . . . . . . . . 616.3.3 Comments on the results . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.4 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A Derivation of potential temperature. 63

B Derivation of the wet equivalent potential temperature 65

C Details of the time consumption of several parts of the LPDM 67

Bibliography 69

6

Chapter 1

Introduction

Throughout history, mankind has always felt the need to predict the weather. Until the20th century, the prediction of the weather was dominated by experience. People knew, bylooking at the sky, whether to expect rain or sunshine. In the 18th and19th century, me-teorological research started to become more and more scientific. People tried to developequations that describe the motion of a fluid. The first equation for an ideal fluid was al-ready introduced in 1755, by Euler. This equation did not incorporate the effect of viscosity.The person who included the viscosity was Claude Navier, in 1822. The equations that areused nowadays were presented in 1845, by George Stokes. At that time, it was impossibleto solve the so called Navier-Stokes equations. People who did research on the equations,could only work on the mathematics, in order to get towards a solution of the equations.With their work and with the invention of the computer it became possible to solve the dis-cretized equations numerically. Performance was poor in the beginning, but, as computersbecame faster and faster, the accuracy of the meteorological models was increasing almostcontinuously.

1.1 The need for parameterization

In atmospheric modeling, the distinction can be made between global weather predictionsand local weather predictions (usually called a General Circulation Model, GCM). The localpredictions are conducted on a fine grid and have limited durability. On the opposite, thelarge scale models have a coarse grid and the simulation time is much larger than the fewdays of the local weather predictions. Depending on the goal, accuracy varies from mod-erate to poor. The grid of a global simulation has nowadays come to in the order of 10 km.With this grid size, it is impossible to account for any effect that has a scale that is smallerthan 10 km.

1.2 Shallow cumulus clouds

Although there are numerous kinds of clouds, in this thesis only one kind of cloud is in-vestigated. Shallow cumulus clouds are typically up to 1 km wide, while they can reachheights of 1-3 km. They have a cloud cover of under 20%, which means that the biggest partof the sky is clear blue sky. Shallow cumulus clouds are therefore also called ’fair weather’cumulus. Since these clouds are relatively small, they are not resolvable in a GCM. Theincorporation of this type of clouds is very important, because these clouds tend to react

7

1.3 Large Eddy Simulations

strongly to a slight change in temperature. Since clouds play a crucial role in the energy bal-ance in climate models, the effect of even the smallest clouds has to be included. The way tosolve this problem, is to parameterize the clouds. In order to improve this parameterization,knowledge is needed about the lifecycle of clouds, about the way dry air is entrained into acloud, about the reflectivity of a cloud and many other properties of a cloud that can haveinfluence on the GCM.

1.3 Large Eddy Simulations

There are two ways to increase knowledge of clouds. The first way is by performing mea-surements. These measurements can be performed by releasing balloons and by flyingthrough clouds by aircraft, but in the last decades more and more measurements are per-formed by using satellite, radar, lidar, etcetera. The properties that are measured are temper-ature, density or moisture profiles; cloud droplet distributions, velocities and so on. Theseprofiles can be used as initial conditions for numerical simulations or as a reference for theoutcomes of simulations. This is the second method for the investigation of shallow cumu-lus clouds. In a Large Eddy Simulation (LES), the lowest few kilometers of the atmosphereare modelled. The grid size of the LES is in the order of 25×25×25 m. This usually is enoughto resolve most of the turbulence in the domain and it is good enough to resolve the largestclouds.

1.4 Motivation for Lagrangian statistics

In the investigation of clouds with LES, there are two problems that emerge. First theamount of data generated by the LES can be enormous, making research slow and (com-putationally) difficult. The second and for this thesis most important problem is that it isvery difficult to see details in the movement of air. A way to overcome these problemsand to extend the possibilities of the research, is to develop a Lagrangian Particle Disper-sion Model (LPDM) that has massless particles traveling through the domain. With sucha model one can, for example, look at a cloud and determine where the particles enteredand exited the cloud. This could answer questions about the entrainment of dry air into acloud. Another important field of research is the dispersion of pollutants in the atmosphere.An LPDM seems to be a natural way to investigate the dispersive properties of the atmo-sphere. Lamb [19] was the first to incorporate such a particle model into an LES in 1978. Hecompared his results to the experimental results of Willis and Deardorff [38]. Others havesuccessfully followed this approach: e.g. Mason [21], Weil et al. [37], Gopalakrishnan andAvissar [10], Kim et al. [15] and Uliasz and Sorbjan [35].

1.5 The goal of the research

The goal of this thesis is to develop, implement, test and use an LPDM. The starting pointis the resolved velocity field, as given by the LES. If there is a finite number of masslessparticles that is released into the domain, then these particles can be tracked in time, bymoving the particles with the resolved velocities. To improve the results, a subgrid schemeis added, based on the results of Weil et al. [37] and Thomson [34]. The LPDM will be used to

8

1. Introduction

look at the entrainment of dry air into a cloud, by means of an analysis presented by Paluch[22].

1.6 Structure of the report

The first chapter of this report gives a short introduction into the subject of research, alongwith the background of the research. In chapter 2 more details are given about the atmo-sphere and the LES. In this chapter the necessary variables and equations will also be treated.After this, chapter 3 will introduce the LPDM, as it is derived and implemented in the LES.After the implementation, in chapter 4 the performance and validity of the LPDM will bediscussed. Chapter 5 will deal with an application of the LPDM to cumulus clouds. At theend, some conclusions will be presented, together with some recommendations for futureresearch.

9

1.6 Structure of the report

10

Chapter 2

The Atmospheric Boundary Layer

The atmosphere surrounding the earth can be divided into many layers (see figure 2.1). Thisresearch will focus on the lowest few kilometers of the atmosphere, called the AtmosphericBoundary Layer (ABL). Together with the free troposphere, the ABL makes up the tropo-sphere. The ABL couples the effects of the surface to the upper regions and is therefore veryimportant in weather prediction, pollution dispersion and many other topics. More infor-mation on the ABL can be found in e.g. Stull [33], Rogers and Yau [27] or Garratt [9].

Figure 2.1: An overview of the layers of the atmosphere (Picture taken from kennislink.nl)

From the surface upwards, the temperature is decreasing with height. The top of the ABL isdetermined by a strong increase in the temperature, called the inversion. Above the inver-sion is the free troposphere where the temperature is again slowly decreasing with height.

11

2.1 Definitions of meteorological quantities

The convective ABL is formed because the sun heats the surface. At places where the air iswarmer than the surrounding air, the air will rise. If there is water vapor in the air, a tur-bulent updraft can cause the rising water vapor to condensate, thus forming clouds. A wayto determine whether a buoyancy driven flow is turbulent or not, is to look at the Rayleighnumber (Ra). This number is defined as:

Ra =gρα∆Td3

kη. (2.1)

Here g is the gravitational acceleration, ρ is the density of air, α is the thermal conductivity,∆T is a temperature difference on a length scale d, k is the thermal diffusivity and η is thedynamic viscosity. In the atmosphere, Ra is in the order of 1016, so flow will be turbulent.This can also be concluded by looking at the Reynolds number (Re):

Re =ρvd

η, (2.2)

where v is a characteristic velocity. For atmospheric conditions, a Reynolds number of 107

can be expected. During the night there is a similar process going on, however with op-posite result. Because of the radiative cooling of the surface, the air close to the surface iscooler than its surrounding, so the ABL is stabilized. This chapter will first introduce somequantities that are commonly used in meteorological science. When this is done, the forma-tion of clouds can be explained. After this, the governing equations that are used in the LESwill be given. Then the subgrid scheme, as implemented in the LES, will be treated. Thisis followed by two case descriptions and finally, there will be some extra information aboutthe LES.

2.1 Definitions of meteorological quantities

It makes sense to express the variables used in the model into quantities that are conservedfor adiabatic processes. In the warm ABL, there is no ice present. The relative water contentcan thus be defined as the sum of the relative amount of liquid water and the relative amountof vapor, present in the air:

qt = ql + qv, (2.3)

where the relative amount represents a mass fraction. In order to derive the equation of stateof the atmosphere, first the equation of state of dry air is given:

p = ρRdT, (2.4)

where Rd is the specific gas constant for dry air. This equation is known as the ideal gas law.It states that only two out of the three variables (p, T, ρ) are independent. For adiabatic pro-cesses, the temperature is not a conserved variable. It needs to be compensated for pressurechanges. This effect can be eliminated by introducing the new, conserved variable, calledthe potential temperature:

θ = T

(p

p0

)−

Rdcp

. (2.5)

12

2. The Atmospheric Boundary Layer

In this equation cp is the specific heat of dry air. The derivation of θ can be found in ap-pendix A. θ can be seen as the temperature that a parcel of air would have if it was broughtadiabatically to the reference pressure p0, in this thesis taken to be 105Pa. If the air containswater vapor, there is another correction necessary to include the effects of phase changes.The new variable is the liquid potential temperature θl.

θl ≈ θ − L

cp

(p

p0

)−

Rdcp

ql, (2.6)

where L is the latent heat of vaporization. Now both θl and qt are conserved variables foradiabatic processes. Since the equation of state of dry air is no longer valid if there is waterpresent, an equation of state for moist air has to be given:

p ≡ ρRdTv, (2.7)

where Tv is the virtual temperature, defined as follows:

Tv =

[1 −

(1 − 1

ǫ

)qv − ql

]T. (2.8)

In this equation, ǫ is the ratio of Rd and Rv. By introducing the virtual temperature, itis possible to keep working with the dry gas constant, instead of a changing ’constant’.Furthermore, the effects of moisture on density can be captured in the introduction of thevirtual potential temperature:

θv = Tv

(p

p0

)−

Rdcp

, (2.9)

Finally, the lapse rate is also of importance in order to determine whether a parcel of air willmove upwards or downwards, if it were to be released at a certain height:

Γθv= −dθv

dz. (2.10)

The overline denotes a horizontal average.

2.2 Formation of shallow cumulus clouds

In this section, the physics behind the formation of clouds is given. On a sunny day, the sunheats up the surface and water evaporates. The surface warms up the air at the bottom ofthe ABL, so the warmed up air will move upward. In figure 2.2 there is a layer where thelapse rate of the dry air is bigger than the lapse rate of the environment. This layer is un-stable and is therefore called the Convective Boundary Layer (CBL). This layer is turbulentuntil the ’level of neutral buoyancy’ is reached. Above this level, the parcel of air that wasmoving upwards has negative buoyancy and is thus forced back into the CBL. If the par-cel has enough momentum, then the parcel may overshoot this level and the parcel mightreach the lifting condensation level. At this level the water vapor starts to condensate, thusforming a cloud. This happens because, during the rise of the parcel, the environmentaltemperature (T ) has been gradually decreasing with height. In a condensation process, heatis released. If there is enough water present, this process can give the parcel of air enough

13

2.3 The governing equations

Figure 2.2: A schematic view on how shallow cumulus clouds are formed.

buoyancy to become once again positively buoyant. The level at which the parcel starts tobecome positively buoyant is called the level of free convection. From this level on, if thereis enough water, the parcel of air will rise all the way to the top of the ABL. Above the levelof zero buoyancy the parcel of air will become negatively buoyant again.

2.3 The governing equations

The equations that need to be solved are: the Navier-Stokes equations for the conservationof momentum, the conservation laws for heat, moisture and mass and the equation of state.The variables that are involved are: the three components of the velocity −→u = (u, v,w), theliquid water potential temperature θl and the total specific humidity qt. In the coordinatesystem used here, the x and y-directions make up a horizontal plane, whereas the z-directionis the vertical direction. For a complete description of the following equations, see e.g. Stull[33] or Garratt [9]. First, the Navier-Stokes equations for incompressible Newtonian fluidsare given:

∂ui

∂t+ uj

∂ui

∂xj= Xi −

1

ρ

∂p

∂xi+ ν

∂2ui

∂x2j

; i, j ∈ [1, 3] . (2.11)

In this equation it is assumed that the density is constant. On the right hand side Xi is thei-component of the external forces acting on the system. The second term is the pressuregradient force and the last term accounts for the viscous stress, with ν the kinematic viscos-ity. Since the Reynolds number is very large (≈ 107), in the remainder of this research theviscous stress term is neglected. The external forces include gravity and coriolis forces:

Xi = −δi3g + ǫij3fcuj + Fi. (2.12)

14


Here δi3 is the Kronecker delta, g is the gravitational acceleration, ǫij3 is the alternating unittensor, fc is the Coriolis parameter and Fi stands for any remaining external forces. In thetensors Einstein’s summation notation is used.

The Boussinesq approximation A further assumption that has to be made is called theBoussinesq approximation. This means that density fluctuations from the reference stateare neglected in all terms of 2.11, except the external buoyancy forcing. This gives for thevertical component:

∂w

∂t+ uj

∂w

∂xj= g

θ′vΘ0

v

− 1

ρ0

∂p′

∂z+ Fz. (2.13)

Here the superscript ′ stands for a perturbation from the reference state of hydrostatic equi-librium, Θ0

v is a reference value of θv. It is assumed that perturbations of the pressure aremuch smaller than temperature or density fluctuations. The equations for the conservedvariables qt and θl are:

∂qt

∂t+ uj

∂qt

∂xj= Fqt , (2.14)

∂θl

∂t+ uj

∂θl

∂xj= Fθl

. (2.15)

In these equations, Fqt represents the sources and sinks of moisture that can be present inthe domain. An example is precipitation. Remember that in phase changes qt is conserved,so they are not in this term. Similar to this, Fθl

contains the heating and cooling forcings,such as results from radiative flux divergence. The equations that make the set of governingequations complete are the mass conservation law:

∂ui

∂xi= 0 (2.16)

and the equation of state, 2.7.

2.4 Large Eddy Simulations of the ABL

This thesis uses the method of LES to solve the equations presented in section 2.3. Sincethese equations can not be solved directly, because that would require too much computerpower, further approximations have to be made. The equations described in the previoussection, will be filtered. This means that all variables are divided in a filtered part and afluctuating, subfilter part.

φ = φ + φ′. (2.17)

The widetilde denotes a filtered variable and the accent denotes a deviation from this value.After this filtering operation, the variables are sampled, since the equations can only besolved for a finite number of grid points. The impact of the unresolved motions needs to bemodelled. The names subgrid and subfilter lie close together in meaning and are often usedinterchangeably. In the remainder of this thesis the name subgrid will be used. The subgrid

15

2.5 Subgrid scale model

model used in the LES is the diffusive-type model of section 2.5. It is possible to transformequation 2.11 into an equation for the filtered variables:

∂ui

∂t+ uj

∂ui

∂xj= −δi3g

θ′′vΘ0

v

+ ǫij3fcuj + Fi −1

ρ0

∂p

∂xi−

∂u′

iu′

j

∂xj. (2.18)

This equation can be used in a model, describing the atmosphere. The last term in theequation represents the impact of subgrid fluctuations on the mean variables and needs tobe modeled. The LES-equations for the conserved variables qt and θl are:

∂qt

∂t+ uj

∂qt

∂xj= Fqt −

∂u′

jq′

t

∂xj, (2.19)

∂θl

∂t+ uj

∂θl

∂xj= Fθl

−∂u′

jθ′

l

∂xj. (2.20)

The last term of these equations also needs to be modeled. The mass conservation law inBoussinesq approximation becomes:

∂ui

∂xi= 0. (2.21)

2.5 Subgrid scale model

Unfortunately, the complete set of equations, as described in the previous section, can not besolved yet. This is because of the higher order terms in equations 2.18, 2.19 and 2.20. Theseequations contain an average over the product of two perturbations. This (co)variance iscalled a second order moment. All equations that are to predict a certain moment, mustcontain higher order moments. This problem is known as the closure problem. The wayto handle this problem is to parameterize the influences, that subgrid terms have on theresolved variables. For equation 2.18, the model of the turbulent flux looks like this:

u′

iu′

j = −Km

(∂ui

∂xj+

∂uj

∂xi

). (2.22)

The higher order moment is now replaced by the local gradient of the mean variable. Theclosure model acts as a diffusion process. The eddy viscosity Km is unknown and has to bemodelled:

Km = Ceλ√

e. (2.23)

In this equation, Ce is a function of the Kolmogorov constant. λ is a mixing length scale andis for now assumed to be proportional to the grid size of the domain:

∆ = (∆x∆y∆z)1

3 , (2.24)

where e is the subgrid turbulent kinetic energy (TKE), defined as:

e =1

2

∑

i

u′2i . (2.25)

16


Since u′

i is not known, an equation for the subgrid TKE must be solved:

∂e1/2

∂t+ uj

∂e1/2

∂xj=

1

2e1/2

(−u′

iu′

j

∂ui

∂xj+

g

Θ0w′θ′v

)− ∂

∂xj

(2Km

∂e1/2

∂xj

)− ǫ

2e1/2. (2.26)

Because in other equations e1/2 is used instead of e, the equation is written in such a way thate1/2 is solved. In equations 2.19 and 2.20, the eddy diffusivity Kφ is defined and modelledas follows:

u′

iφ′ = −Kφ

∂φ

∂xi. (2.27)

In this equation, φ is either θl or qt. The eddy diffusivities for the conserved variables arerelated to Km by the turbulence Prandtl number Pr:

Kθl= Kqt = Pr−1Km. (2.28)

The Prandtl number is an approximation of the ratio of momentum diffusivity and thermaldiffusivity. In the LES, a Prandtl number is used of 1

3 . Furthermore, the dissipation rate ǫ

can be modelled by:

ǫ =cǫe

2/3

λ, (2.29)

where cǫ is given by:

cǫ = cǫ,1 + cǫ,2λ

∆. (2.30)

For the λ in these equations the grid size ∆ (2.24) , is used in case of turbulence. For thestable flow (∂θv

∂z > 0) a correction must be made, so λ is taken to be:

λ = min

(∆, cN

e1/2

N

), (2.31)

N2 =g

Θ0

∂θv

∂z.

The N2 mentioned above is called the Brunt-Vaisala frequency. This frequency is a measurefor the strength of the stratification. In the LES used in this thesis, cǫ,1 = 0.19, cǫ,2 = 0.51 andcN = 0.76.

2.6 Case description

In this thesis two cases are used. One is the cloudy BOMEX-case, the other is a clear CBL-case. The cases differ, amongst others, in the initial and boundary conditions of the LES andin the large scale forcing that is applied. Now more details will be given about the cases.

17

2.7 Boundary and initial conditions

BOMEX The Barbados Oceanographic and Meteorological EXperiment was conducted in 1969in the ABL above the ocean, east of Barbados. In this experiment, a fleet of 5 ships measuredthe surface fluxes, while every 90 minutes a sonde was released to measure the verticalprofiles of temperature, moisture and wind. The results of these measurements are used asan initial condition for the LES, see Siebesma et al. [29]. Here the domain size was 6.4 × 6.4× 3.2 km. The grid resolution varied from 64× 64× 80 points, with a timestep of 2 secondsin test cases, to 256 × 256 × 160, with a timestep of 1 second in detailed simulations. Adry well-mixed layer of about 500 m height is topped by a conditionally unstable cloudlayer, which in turn is capped by a stable inversion at 2000 m. A central differencing schemeis used to calculate the advection of the scalar quantities. Because the BOMEX case has adominating wind in one direction, a Gallilei transformation is performed. This means thatthe mean wind is subtracted from the local wind velocities.

CBL The domain size of the simulations of a Convective Boundary Layer were somewhatsmaller. The size was 3.2 × 3.2 × 1.2 km, with a resolution of 32 × 32 × 48. The timestepis taken to be 1 s. In this case no clouds can be formed, since there is not enough water inthe air, so only the dry, well-mixed layer is present. This layer is about 775 m high and isconfined by an inversion with a strength of about 5 K. In this case the so called κ-schemeis used to calculate the advection of the scalars. More information on this scheme can befound in Hundsdorfer et al. [11]. The relative simplicity of the CBL should make it a goodcandidate for test applications.

2.7 Boundary and initial conditions

The boundary and initial conditions of the LES are as follows:

at the bottom At the bottom of the LES, no-slip conditions for velocities are applied. Thismeans that u = v = w = 0. The total specific humidity flux and the liquid water potentialtemperature flux are set to a ground level value: φqt,surface and φθl,surface. Values that aredefined at the cell centers need to be extrapolated towards the bottom of the domain.

At the top At the top of the LES, a free surface boundary condition is applied. The verticalderivatives of the horizontal velocities are set to zero, together with the vertical velocity andthe sgs-TKE. This gives ∂eu

∂z = 0, ∂ev∂z = 0, w = 0 and e = 0. For the total specific humidity and

the liquid water potential temperature, the vertical derivative is set to a constant: ∂qt

∂z = c1,∂θl

∂z = c2. Values that are defined at the cell centers need to be extrapolated towards the topof the domain.

At the sides At the sides of the LES, periodic boundary conditions are applied, for allvariables.

At the start The begin conditions are as mentioned in the previous section. All variablesare set planewise with values derived from experiments. Because the initial state of thesystem is completely uniform at every height, nothing would happen if the surface fluxeswere turned on. The heat and water that are added at the surface would have to be trans-ported upwards by diffusion, because there can only be convection (in a closed system) if

18


there is an initial perturbation of the system. In practice this would mean that the computeraccuracy would be dominating in the emergence of turbulence. This would imply a strongdependency on the used computer. In order to prevent this undesired effect, small randomperturbations are added in both qt and θl. Since the discrete starting values are not an exactphysical description of the initial state of the system, the system needs some time to stabilizeitself after a startup. Usually the first three hours of simulation are neglected.

2.8 Remaining properties

The timestep In order to choose a timestep, the criterion is used that the Courant number(Co) should be smaller than 1. This dimensionless number is defined as follows:

Co =vmaxdt

min(∆x,∆y,∆z). (2.32)

In both cases, the velocities are always smaller than 10 ms−1. For a smallest grid size of 20m (in vertical direction), the timestep should be smaller than 2 seconds, hence a timestep of1 second was used in the detailed simulations.

The parallelization To speed up the simulation, the LES-code is parallelized so that therecan be multiple processors working on a simulation. This is done by dividing the domainof the LES into n parts, where n stands for the number of processors (usually a power of 2).The division of the domain takes place in the y-direction. The consequence of the division isthat variables at the edge of the sub-domains should be communicated after every timestepof the LES.

Staggered grid The LES has a staggered grid implemented in order to avoid oscillatoryeffects, that appear if every variable is calculated at the same position, at regular distances.A staggered grid means that a velocity is defined at the face of a gridbox. The other variables,like the temperature, pressure and water content, are defined in the center of the gridbox.

19

2.8 Remaining properties

20

Chapter 3

Lagrangian Particle Dispersion Model

The usage of Lagrangian Stochastic models is very common in atmospheric science. Manyresearch groups from all over the world are (or were) using this kind of model. In sucha model the trajectories of massless particles are calculated by means of a random-walkscheme. This means that the Navier-Stokes equations from equation 2.11 are not solved,but the particle motion is described by using a statistical description of the random velocityfield. See for examples of these atmospheric stochastic models e.g. Wilson and Sawford [39],Kos et al. [18], Rao [24], Franzese et al. [8], Stohl and Thomson [32], Warland and Thurtell[36], Kljun et al. [16], Stohl et al. [31], k. Luhar and Britter [13], Reynolds and Iacono [25],Borgas and Sawford [3], Sawford and Yeung [28], Das and Durbin [6] and Du et al. [7]. Theprinciple has also been applied in oceanographic research by Brickman and Smith [4].

Another way of investigation of turbulent flows, is the application of particle tracking. Nowthe particles are moved by a velocity field that is calculated by solving the Navier-Stokesequations. Research on particle tracking has been performed by, amongst others: Kenwrightand Lane [14], Darmofal and Haimes [5] and Lane [20]. A combination of Direct NumericalSimulation (DNS) and particle tracking has been investigated by Yeung and Pope [40] andKontomaris et al. [17].Unfortunately, DNS is not possible in the atmosphere, because of thehigh Reynolds number of ∼ 107.

3.1 Particle tracking in an LES

The main goal in this thesis is to implement a Lagrangian Particle Dispersion Model (LPDM)into the LES in order to calculate the particle trajectories online. This idea is also applied by,amongst others: Mason [21], Weil et al. [37], Gopalakrishnan and Avissar [10], Kim et al.[15] and Uliasz and Sorbjan [35]. The particles are massless and infinitesimally small. Theparticle subroutine will run parallel to the LES, so that at every moment, it is possible to getan output from the particle scheme, together with the output of the LES. It is also necessaryto parallelize the particle scheme to be able to track as many particles as possible, withoutslowing the LES down for too much. The equation that needs to be solved in order to tracka particle is the following:

dxi,p

dt= ui(

−→xp; t) (3.1)

Where ui are the resolved velocities in the x-, y- and z-direction and −→xp are the coordinatesof the particle. With the resolved velocities at every gridpoint at hand, from the LES, there

21

3.2 The interpolation scheme

are four important points of attention. These are:

1. spatial interpolation scheme

2. integration scheme

3. subgrid scheme

4. boundary handling scheme

At every timestep, the LPDM obtains the discrete, resolved velocities in three directions fromthe LES. After this, the LPDM has to interpolate these velocities for each particle, becausethe particles can move freely between the gridpoints. Then the new positions of the particlesare calculated using the integration scheme. Since the LES only gives the resolved velocities,there is need for an extra scheme to account for the influence of subgrid turbulence. Theinput velocities for the integration scheme are divided into a resolved and a subgrid part.At the end of the LPDM, the boundary handling scheme makes sure that the particles staywithin the domain of the LES.

3.2 The interpolation scheme

A point of attention in interpolation is that the LES employs a staggered grid. This meansthat a velocity is defined at all faces. The other variables, like the temperature, pressure andwater content, are defined in the center of the gridbox, see figure 3.1. Although this lookscomplicated, it turns out that for 1 velocity component, a staggered grid turns into a shiftedregular grid.

Figure 3.1: An illustration of a staggered grid in 2D.

Although there are many sophisticated interpolation schemes available from literature, in

22

3. Lagrangian Particle Dispersion Model

Figure 3.2: An illustration of tri-linear interpolation, as explained in equation 3.2.

the LPDM, a simple tri-linear interpolation scheme is used:

up = (1 − dx) (1 − dy) (1 − dz) uA + (3.2)

dx (1 − dy) (1 − dz) uB +

dx dy (1 − dz) uC +

(1 − dx) dy (1 − dz) uD +

(1 − dx) (1 − dy) dz uE +

dx (1 − dy) dz uF +

dx dy dz uG +

(1 − dx) dy dz uH .

Here the uX are defined as shown in figure 3.2. There can be up to 10 million particles inthe domain and if the simulation lasts for several hours, with a timestep of 2 seconds, then acomplicated interpolation scheme like cubic splines, might take too much computation time,compared to the time needed to compute a timestep in the LES. Some tests will be performedin order to see whether the used interpolation scheme is sufficient in accuracy. These testswill involve the calculation of horizontally averaged statistics, by putting particles at a rigidposition at either the center, or at an edge of a gridbox. Since the particles can not move, anydifference in statistics will then be caused by the interpolation scheme. The results can befound in the next chapter.

3.3 The integration scheme

With the right velocities at hand, it is possible to make an estimate for the particle positionafter one timestep. There are numerous ways to do so, the simplest being Euler’s integrationscheme:

xi,n+1 = xi,n + vi(−→x n, t)dt (3.3)

23

3.3 The integration scheme

Now the 1st order Euler-scheme is not the best scheme available. There are many different,higher order schemes that will allow the particles to follow the local streamlines in a betterway. A few commonly used schemes are:

x∗

i,n+1 = xi,n + vi(−→x n, t)dt (3.4)

xi,n+1 = xi,n +1

2

(vi(

−→x n, t) + vi(−→x∗

n, t))

dt

This method is called Heun’s method, also known as a 2nd order Runge-Kutta method.Another popular, second order method is the Adams-Bashfort scheme:

xi,n+1 = xi,n +

[3

2vi(

−→xn) − 1

2vi(

−−→xn−1)

]dt (3.5)

Finally, the 4th order Runge-Kutta scheme has to be mentioned, as it is the most common4th order interpolation scheme:

k1 = vi(−→x n, t)dt, (3.6)

k2 = vi(−→x n +

1

2k1, t +

1

2dt)dt, (3.7)

k3 = vi(−→x n +

1

2k2, t +

1

2dt)dt,

k4 = vi(−→x n + k3, t + dt)dt,

xi,n+1 = xi,n +1

6(k1 + 2k2 + 2k3 + k4),

Although this thesis does not go into the details of the different integration schemes, a fewgeneral comments on the integration schemes can be formulated:

• speed

• accuracy

• stability

• applicability

speed The integration scheme has to be used at every timestep of the particle subroutineand for every particle. Since the number of particles is usually very high (>100.000) in orderto maintain statistical convergence, it becomes very important for any scheme involved intracking the particles to be as fast as possible. As an example: a detailed simulation carried1,3 million particles and the integration scheme was called 3 times at every timestep ofthe LPDM, while the simulation lasted for 15 hours. This example gives a total number ofloops through the integration scheme of about 1011. The demand for a fast algorithm pointstowards the simple Euler scheme. A higher order scheme will take more time. The goal isto have the LPDM using no more than 50% of the total computation time.

accuracy In general higher order integration schemes will be more accurate. This makesit very likely that the fast 1st order Euler scheme will give poor results. The demand foraccuracy is conflicting with the demand for speed. Fortunately there is a limit on this. Sincethe velocities, that are the input of the integration scheme, are filtered in the LES, there areno rapid changes in the velocity possible. In this thesis, some tests are performed to see

24


whether the 2nd order Adams-Bashfort scheme is indeed outperforming the 1st order Eulerscheme and whether it is good enough to be used in production runs. If it turns out thatthe 2nd order Adams-Bashfort scheme is performing better than the 1st order Euler scheme,then it might become necessary to implement a 4th order scheme (e.g. Runge-Kutta), to seeif that would further increase the performance of the LPDM. The tests will be described inthe next chapter.

stability In order to make the LPDM unstable, the velocity that is used to advect the parti-cles must become infinite. In the LPDM this velocity is interpolated from the velocities thatare resolved in the LES. This means that if the LES is stable, then the LPDM is stable. In thisthesis the LES is assumed to be stable, so there were no problems regarding the stability ofthe integration schemes.

applicability This requirement concerns the discretization of the velocity field in time.Some integration schemes require more than 1 velocity field (e.g. Heun, 4th order Runge-Kutta) per timestep. These intermediate velocity fields are not present in LES, unless thetimestep of the particle scheme is chosen twice the timestep of the LES. But then it is highlyquestionable if the increased accuracy of the 2nd order integration method, will balance thedecreased performance due to the larger timestep.

In the LPDM, there are 3 options for the integration scheme available. These are: the 1storder Euler scheme, the 2nd order Heun scheme and the 2nd order Adams-Bashfort scheme.It is now possible to choose per simulation, which scheme will be used. Because of the mod-ular approach, it is very easy to add more integration schemes into the subroutine in thefuture.

3.4 The subgrid scheme

Although there is already a subgrid scheme present in the LES, a second subgrid scheme isneeded for the LPDM. This is due to the fact that the input to the particle subroutine con-sists of resolved velocities, that are unable to fully describe the particle motions. Especiallyat the bottom of the domain and at the inversion, there is a lot of subgrid turbulence thatis not resolved by the LES. A criterion for an subgrid scheme is that particles that are uni-formly spread in the domain, should remain uniformly distributed. In order to meet thiswell-mixed condition, a Langevin model is adopted based on the research of Thomson [34].An adaptation towards LES has been made by Weil et al. [37]. For more information onLangevin equations and solving them by means of solving the Fokker-Planck equation, seee.g. Pope [23].

The equation that needs to be solved is 3.1. In this equation ui stands for the filtered ve-locity. The velocity that is used to advect a particle, is composed of a resolved part and afluctuating, subgrid part. This will lead to an expansion of equation 3.1:

dxi,p

dt= ui(

−→xp; t) + u′

i(−→xp; t), (3.8)

where the ui are resolved by the LES. The random subgrid velocity components u′

i are as-sumed to behave as a Markov process. This means that the velocity at the next timestep is

25

3.4 The subgrid scheme

a function of the velocity at the present timestep. From theory it follows that the evolutionof u′

i can then be described by the following stochastic differential equations (also called aLangevin equation):

du′

i = ai(−→x ,−→u , t)dt + bij(

−→x ,−→u , t)dξj , (3.9)

where a and b are functions of −→x , −→u and t. The total velocity −→u is the sum of the resolvedvelocity and the subgrid velocity. The dξj is a component of a Gaussian white noise. The bij

term determines the distribution of the random velocity forcing and is chosen to be consis-tent with the Lagrangian structure function in the inertial subrange:

bij = (C0ǫ)1/2. (3.10)

The C0 is the Langevin-model constant (= 4 ± 2; Thomson [34], 4.3 ± 0.3; Rizza et al. [26]), ǫ isthe dissipation rate. The density function of the phase-space (i.e. (−→x ,−→u )-space) distributionof the particles will be denoted by f(−→x ,−→u , t). For a Gaussian random forcing, the ai termhas to satisfy the Fokker-Planck equation. This equation governs the evolution of the La-grangian PDF (fL(−→x ,−→u , t)) in phase space and gives the solution to a stochastic differentialequation:

∂fL

∂t=

1

TL

∂(−→u fL)

∂−→u+

σ2

TL

∂2fL

∂−→u 2. (3.11)

Here −→u is the total velocity and TL is the Lagrangian time scale. For a well-mixed distri-bution, the Lagrangian PDF should be equal to the Eulerian PDF, since if the particles arewell-mixed, then the particles completely fill the domain in both cases, thus giving iden-tical results. Thomson argues, that the ai term in equation 3.9 can be calculated from theFokker-Planck equation using the Eulerian PDF. More information about the derivation canbe found in Thomson [34]. In Weil et al. [37], the following equation for the subgrid veloci-ties is derived:

du′

i = −fsC0ǫu′

i

2σ2sgs

dt +1

2

(1

σ2sgs

dσ2sgs

dtu′

i +∂σ2

sgs

∂xi

)dt + (fsC0ǫ)

1/2dξi. (3.12)

Here fs is a coefficient that takes into account that there is no need to model the entire ve-locity, since there is already a part resolved by the LES. The variance of the subgrid velocity,σ2

sgs is equal to 23e, where e is the subgrid turbulent kinetic energy. A parameterization of fs

is given by:

fs =σ2

sgs

σ2av + σ2

sgs

, (3.13)

where the overline means an average over a horizontal plane. The σ2av is the horizontally

averaged resolved variance:

σ2av =

1

3

∑

i

(ui − ui)2. (3.14)

The form of equation 3.12 is equal to the form of a Langevin equation. The first term is aviscous term that tends to diminish the velocity fluctuations. The second term relates theparticle velocity to the subgrid TKE. This relation is important, since the subgrid TKE is ameasure for the amount of unresolved turbulence. Finally, the last term gives a randomfluctuation similar to Brownian motion.

26


3.5 The boundary handling scheme

The boundary conditions for the particles are related to the boundary conditions of the LES.Relevant for the particles are:

• at bottom: no-slip conditions, u = v = w = 0

• at top : free surface, ∂eu∂z = 0, ∂ev

∂z = 0, w = 0

• at sides : periodic boundary conditions

Near the bottom and at the top, linear interpolation will take place between nearest grid-points and the boundary. Furthermore, at the bottom and at the top of the domain, particlesthat hit the boundary will be perfectly reflected. Note that the possibility of massless parti-cles hitting the boundaries is only possible in combination with a very large timestep or withthe use of the subgrid model. The no-slip conditions at the bottom, gives rise to relativelylarge gradients in the lowest grid boxes, since any velocity goes from any value to zero inthe lowest grid box. The main goal of the top boundary condition is to prevent a particlefrom moving outside the domain.

Like in the LES, the parallelization makes the handling of the boundary conditions in they-direction a little bit different from that in the x-direction. If a particle moves across theupper boundary in x-direction, then it comes in at the lower boundary again. In the y-direction, the particle that moves across the upper boundary, is moved to the next processorand comes in from the lower boundary again. In order to cope with the problem of individ-ual processors having a changing number of particles, a dynamic list structure was used forparticles at every processor. With this so called linked list the movement of particles betweenprocessors became easier. A particle crossing the y-boundary will be removed from the listof the old processor and then added to the list of the new processor. Every list contains asmany records as there are particles inside the domain of the processor, while every recordconsists of as many entries as wished. The boundary handling scheme also has to be calledfor intermediate results (necessary for some interpolation schemes, e.g. Heun), because anew velocity can only be calculated, if a particle is inside the domain of the LES.

27

3.5 The boundary handling scheme

28

Chapter 4

Validation of the LPDM

After the implementation of the particle routine, the question remains whether the particlesdo accurately follow the local streamlines. It is not obvious that a complicated scheme asdescribed in the previous chapter, will give good results. Several criteria can be formulatedin order to get a grip on the accuracy of the LPDM. These criteria can be divided into threegroups:

1. The particle routine should be stable.

2. Particles that begin uniformly distributed, must stay that way. (This is the well-mixedcondition, as mentioned in section 3.4)

3. If the particles are uniformly distributed, they should give the same statistics in La-grangian way, as they would in an Eulerian frame.

In the remainder of this chapter these criteria will be used as a guideline in order to validatethe LPDM within the framework of the LES. At the end of this chapter, some informationwill be given about the time consumption of the LPDM.

4.1 Stability of the particle scheme

Although this requirement sounds rather simple, it is of the greatest importance, becausethe particle routine would be useless if there were any regions where the solution of theparticle advection scheme would be unstable. Fortunately, the scheme has proven itself tobe stable. Simulations of high resolution, carrying over a million particles had no problemsreaching the end of the simulation. No particles were found having velocities in excess ofthe expected maximum velocity of 8 ms−1. However, when the particle routine was exe-cuted with the CBL initial conditions, there were serious problems regarding the stability ofthe LPDM. It was found that there was a problem in combining the κ-scheme that is usedin CBL simulations, with the sgs-scheme of the LPDM. A κ-scheme is designed to advectscalars (e.g. qt, e), in such a way that the scalar can never become negative. This keeps theLES free from negative amounts of water or negative amounts of sgs-TKE.

The problem can be explained by looking at figure 4.1. In this figure, the line after 1 hourshows that the variable fs (equation 3.13) has a peak above the inversion height. This peakis induced by the fact that the sgs-variances go to zero in a slower way than the resolved-variances, above the inversion. Now the κ-scheme, that is used as an advection scheme

29

4.2 The well-mixed condition

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1

z/z i

(-)

fs (-)

fs after 1 hourfs after 10 hours

Figure 4.1: The fs parameter as a function of height. This parameter determines the fraction of sgs-TKE that is used in calculating the sgs-velocities. The height is normalized with the height of theinversion after 1 hour, zi = 775 m. The height of the inversion after 10 hours is slightly higher.

inside the LES is a dissipative scheme, hence it destroys some of the sgs-variances, so thatthe sgs-variances go to zero faster than they should and thereby the fs parameter goes tozero instead of having a peak. This is shown by the line after 10 hours. The fact that the fs

parameter becomes equal to zero, means that the 1st term and the last term in equation 3.12also become equal to zero. Because the only remaining term makes the velocity grow eachtime it is calculated, the LPDM became unstable. In this thesis the problem was solved bymanually setting fs = 1 for all cases. This means that the LPDM as used in this thesis is posi-tioned in the middle of the models as posed by Weil et al. [37] and Thomson [34]. Thomsonmimics all particle motion by a stochastic process, whereas Weil lets only the part that is notresolved in the LES be determined by a stochastic process. Although this adjustment seemsa bit harsh, the scheme was again found to be stable. However, it may have an influence onthe well-mixed condition or on statistics, derived from the LPDM.

4.2 The well-mixed condition

To check whether the LPDM satisfied the well-mixed condition, as posed by Thomson [34],a BOMEX-simulation was performed, containing over 10,000,000 particles in a domain of256×256×160 grid points. The domain was divided into 160 layers. At every timestep, thenumber of particles in every layer was counted. The simulation without the subgrid schemeresulted in figure 4.2. As can be seen in the figure, there is a large amount of particles atthe bottom of the domain. This surplus of particles is grows in time to an amount of ap-proximately 50%, where it stabilizes. It is clear that up to a relative height of 0.4 there is aninfluence on the number of particles present at every height. After the implementation ofthe subgrid scheme, the test was repeated. Figure 4.3 shows that only at the bottom thereis a small surplus of particles of about 5%. This behavior is stable in time, so the LPDM isperforming quite well with regard to the well-mixed condition. This is not perfect, but itis sufficient for current use, as the main application of this LES is the bulk of the domain.

30

4. Validation of the LPDM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

z/z i

(-)

# particles / # particlesref (-)

2 hours 4 hours

6 hours

without sgs, after 2 hourswithout sgs, after 4 hourswithout sgs, after 6 hours

Figure 4.2: The relative number of particles as a function of heightas a result of a simulation withoutthe subgrid scheme. The height is normalized with the height of the inversion, zi = 2000 m. Thereference number of particles is 65536 (256 × 256).

It is also believed, that with the addition of a more physical surface handling scheme (e.g.Monin-Obukhov), this deficiency of the LPDM could be further reduced. A Monin-Obukhovscheme would include a logarithmic decay of velocities towards the bottom of the domainif the ABL were neutral. Since the ABL is (in this study) far from neutral, the profiles wouldslightly deviate from the logarithmic curve.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

z/z i

(-)

# particles / # particlesref (-)

with sgs, after 2 hourswith sgs, after 4 hourswith sgs, after 6 hours

Figure 4.3: The relative number of particles as a function of height as a result of a simulation withthe subgrid scheme. The height is normalized with the height of the inversion, zi = 2000 m. Thereference number of particles is 65536 (256 × 256).

31

4.3 Comparison of Eulerian-Lagrangian statistics


When the particles are uniformly distributed and if there are enough particles, then the La-grangian statistics should be equal to the Eulerian statistics. In the LES, several statistics arecalculated in an Eulerian way. These include horizontal slab averages of velocities, variancesand subgrid turbulent kinetic energy. In the following figures, these variables are shown,calculated both in an Eulerian and in a Lagrangian way.

4.3.1 The BOMEX case

velocities The simplest comparison that can be made, is the comparison between the hor-izontally averaged velocities. In figure 4.4 and 4.5 the results of this comparison are shownfor the BOMEX case. It can be seen that the results from the LPDM are in good agreementwith the LES values.

variances Another quantity that can be used to compare the results of the LES with thosefrom the LPDM, are the horizontally averaged, resolved variances. These results are shownin figure 4.6, 4.7 and 4.9. It is clear that the values from the LPDM are a little smaller than thevalues from the LES. This is most likely due to the fact that velocities have to be interpolatedin the LPDM. This effect can be explained by looking at the definition of the variance:

σ2u = (u − u)2 = u2 − u2 (4.1)

The overline means a horizontal average. Now it has just been shown that u, as measuredby the LPDM is equal to the averaged velocity derived from the LES. Thus the differencehas to be caused by the u2 term. Now it is known that it matters whether the interpolationtakes place before or after the squaring. In the LPDM the velocity is first interpolated andthen squared. This gives rise to a slight underestimation of the resolved variances. Apartfrom this, the results from LES and LPDM show a great resemblance.

subgrid scale turbulent kinetic energy A third quantity that can be looked at is the sgs-TKE. The sgs-TKE should be equal to the sum of the three variances of the sub-grid veloci-ties, divided by 2 (see equation 2.25). The results of that comparison are shown in figure 4.9.Also in this case the figure shows a good agreement of LES results and LPDM results.

4.3.2 The CBL case

velocities Also for the CBL case, the comparison is made between the horizontally aver-aged velocities. In figures 4.10 and 4.11 the results are shown. The results from the LPDMshow reasonable agreement with the LES values. The values for the CBL case are more scat-tered. This is due to the fact that the average is closer to zero and the number of particleswas about 6 times smaller than in the BOMEX case.

variances The second quantity that is used to compare the results of the LES with thosefrom the LPDM, are the variances. The results for the CBL-case can be found in figures4.12, 4.13 and 4.15. Here also, the values from the LPDM are a little smaller than the valuesfrom the LES. Here it is shown that this is a result of the interpolation of the velocities inthe LPDM. The values of the LPDM lie between the values of a simulation with the LPDM

32


0

0.2

0.4

0.6

0.8

1

1.2

-8 -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5

z/z i

(-)

velocity (m/s)

EulerianLagrangian, particle SGS onLagrangian, particle SGS off

Figure 4.4: Comparison between the instantaneous horizontally averaged u-velocities as obtainedby standard LES output and the horizontally averaged u-velocities as calculated by the LPDM. Thedifference between the lowest points is caused by the fact that the simulation without the sgs-schemegives more particles at the bottom and hence the average velocity is underestimated. The sample timeis at 2 hour after the restart of the simulation. The restart took place after 4 hours of simulation. Thisis a BOMEX case. The height is normalized with the height of the inversion, zi = 2000 m.

0

0.2

0.4

0.6

0.8

1

1.2

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2

z/z i

(-)

velocity (m/s)


Figure 4.5: Comparison between the instantaneous horizontally averaged v-velocities as obtainedby standard LES output and the horizontally averaged v-velocities as calculated by the LPDM. Thesample time is at 2 hour after the restart of the simulation. The restart took place after 4 hours ofsimulation. This is a BOMEX case. The height is normalized with the height of the inversion, zi =2000 m.

33


0

0.2

0.4

0.6

0.8

1

1.2

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36

z/z i

(-)

variance (m2/s2)

σu2 from LES

σu2 from LPDM with SGS

σu2 from LPDM without SGS

Figure 4.6: Comparison between the instantaneous horizontally averaged u-variances as obtained bystandard LES output and the horizontally averaged u-variances as calculated by the LPDM, with orwithout sgs-scheme. The sample time is at 2 hour after the restart of the simulation. The restart tookplace after 4 hours of simulation. This is a BOMEX case. The height is normalized with the height ofthe inversion, zi = 2000 m.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28

z/z i

(-)

variance (m2/s2)

σv2 from LES

σv2 from LPDM with SGS

σv2 from LPDM without SGS

Figure 4.7: Comparison between the instantaneous horizontally averaged v-variances as obtained bystandard LES output and the horizontally averaged v-variances as calculated by the LPDM, with orwithout sgs-scheme. The sample time is at 2 hour after the restart of the simulation. The restart tookplace after 4 hours of simulation. This is a BOMEX case. The height is normalized with the height ofthe inversion, zi = 2000 m.

34


0

0.2

0.4

0.6

0.8

1

1.2

0 0.04 0.08 0.12 0.16 0.2

z/z i

(-)

variance (m2/s2)

σw2 from LES

σw2 from LPDM with SGS

σw2 from LPDM without SGS

Figure 4.8: Comparison between the instantaneous horizontally averaged w-variances as obtainedby standard LES output and the horizontally averaged w-variances as calculated by the LPDM, withor without sgs-scheme. The sample time is at 2 hour after the restart of the simulation. The restarttook place after 4 hours of simulation. This is a BOMEX case. The height is normalized with theheight of the inversion, zi = 2000 m.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25

z/z i

(-)

mean energy (m2/s2)

sum of sgs-variancesmean sgs-tke

Figure 4.9: Comparison between the instantaneous horizontally averaged sgs-TKE as obtained bystandard LES output and the horizontally averaged variances as calculated by the LPDM. This is aBOMEX case simulation. The sample time is at 2 hour after the restart of the simulation. The restarttook place after 4 hours of simulation. The height is normalized with the height of the inversion, zi

= 2000 m.

35


0

0.2

0.4

0.6

0.8

1

1.2

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

z/z i

(-)

velocity (m/s)


Figure 4.10: Comparison between the instantaneous horizontally averaged u-velocities as obtainedby standard LES output and the horizontally averaged u-velocities as calculated by the LPDM. Thesample time is at 10 hour after the restart of the simulation. The restart took place after 3 hours ofsimulation. This is a CBL case. The height is normalized with the height of the inversion, zi = 775 m.

with non-moving particles at the center of the gridboxes and a simulation with non-movingparticles at the edge of the gridboxes. Apart from this, the results from LES and LPDM showa great resemblance.

subgrid scale turbulent kinetic energy A third quantity that can be looked at is the sgs-TKE. The sgs-TKE should be equal to the sum of the three variances of the sub-grid veloc-ities, divided by 2 (see equation 2.25). The results of that comparison are shown in figure4.15. Also in this case the figure shows a good agreement of LES results and LPDM results.

4.3.3 The vorticity

An interesting quantity in turbulence research is the vorticity of the flow. The vorticity isdefined as:

−→ω =−→∇ ×−→u (4.2)

The vorticity was calculated inside the LPDM and averaged over the entire domain. It wasanticipated that particles, advected without an sgs-scheme would have a lower vorticity,because they would be thrown out easier of the fastest spinning eddies. This numericaleffect is due to the fact that a particle is advected with an integration scheme that has acertain timestep. During this timestep, a particle will travel in a straight line and hence theparticle will be thrown out of the vortex. The results of this experiment are shown in figure4.16. It can be seen that the particles that are advected without an sgs-scheme have a highervorticity than the particles that were advected with the sgs-scheme. This unexpected resultcan be explained by combining figures 4.2 and 4.17. In figure 4.2 it can be seen that, withouta sgs-scheme, there are more particles at the bottom of the LES. Figure 4.17 shows that at thebottom of the domain, the vorticity is highest. So this explains the higher overall vorticityfor the LPDM without sgs-scheme. Furthermore, figure 4.16 shows that the implementation

36


0

0.2

0.4

0.6

0.8

1

1.2

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

z/z i

(-)

velocity (m/s)


Figure 4.11: Comparison between the instantaneous horizontally averaged v-velocities as obtainedby standard LES output and the horizontally averaged v-velocities as calculated by the LPDM. Thesample time is at 10 hour after the restart of the simulation. The restart took place after 3 hours ofsimulation. This is a CBL case. The height is normalized with the height of the inversion, zi = 775 m.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32

z/z i

(-)

variance (m2/s2)

σu2 from LES

σu2 from LPDM with SGS

σu2 from LPDM at cell centers

σu2 from LPDM at cell edges

Figure 4.12: Comparison between the instantaneous horizontally averaged u-variances as obtainedby standard LES output and the horizontally averaged u-variances as calculated by the LPDM. In thisfigure 4 lines are plotted. The first are the averaged variances as obtained from the LES. The secondis obtained by the LPDM with moving particles. The last two are obtained by putting non-movingparticles at either the center or the upper edge of the gridboxes. The sample time is at 10 hour afterthe restart of the simulation. The restart took place after 3 hours of simulation. This is a CBL case.The height is normalized with the height of the inversion, zi = 775 m.

37


0

0.2

0.4

0.6

0.8

1

1.2

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36

z/z i

(-)

variance (m2/s2)

σv2 from LES

σv2 from LPDM with SGS

σv2 from LPDM at cell centers

σv2 from LPDM at cell edges

Figure 4.13: Comparison between the instantaneous horizontally averaged v-variances as obtainedby standard LES output and the horizontally averaged v-variances as calculated by the LPDM. In thisfigure 4 lines are plotted. The first are the averaged variances as obtained from the LES. The secondis obtained by the LPDM with moving particles. The last two are obtained by putting non-movingparticles at either the center or the upper edge of the gridboxes. The sample time is at 10 hour afterthe restart of the simulation. The restart took place after 3 hours of simulation. This is a CBL case.The height is normalized with the height of the inversion, zi = 775 m.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28

z/z i

(-)

variance (m2/s2)

σw2 from LES

σw2 from LPDM with SGS

σw2 from LPDM at cell centers

σw2 from LPDM at cell edges

Figure 4.14: Comparison between the instantaneous horizontally averaged w-variances as obtainedby standard LES output and the horizontally averaged w-variances as calculated by the LPDM. Inthis figure 4 lines are plotted. The first are the averaged variances as obtained from the LES. Thesecond is obtained by the LPDM with moving particles. The last two are obtained by putting non-moving particles at either the center or the upper edge of the gridboxes. The sample time is at 10hour after the restart of the simulation. The restart took place after 3 hours of simulation. This is aCBL case. The height is normalized with the height of the inversion, zi = 775 m.

38


0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12

z/z i

(-)

mean energy (m2/s2)

sum of sgs-variancesmean sgs-tke

Figure 4.15: Comparison between the instantaneous horizontally averaged sgs-TKE as obtained bystandard LES output and the horizontally averaged variances as calculated by the LPDM. The sampletime is at 10 hour after the restart of the simulation. The restart took place after 3 hours of simulation.This is a CBL case simulation. The height is normalized with the height of the inversion, zi = 775 m.

of the sgs-scheme makes the particles distribute more quickly towards a steady state. Ifwe now take a closer look at figure 4.17, then it can be seen that the horizontally averaged,absolute vorticity does not depend on the used integration scheme nor on the usage of thesubgrid scheme. However, the difference in sampling position is evident. The reason for thisdifference, is that, in order to calculate a derivative like ∂u

∂y , there is also a difference in theamount of interpolation that is needed to perform the calculation. At the center of a gridbox,there are more interpolation steps necessary than at the edge. This increased amount ofinterpolations gives rise to a lower averaged vorticity. All this leads to the conclusion thatthe interpolation scheme is a greater source of error, than the integration scheme is.

4.4 Computation time of the LPDM

After the LPDM was implemented and validated, it was possible to measure the time theLPDM spends in a certain subroutine. The results of such a measurement are shown in ap-pendix C. Unfortunately, the measured simulation was so large, that it was impossible tocombine the data of all 32 processors, so only the data from 1 processor is given. Becausethe particles were, more or less, homogeneously spread, this data should be representativefor the total domain. It can be seen that in the simulation that contained over 10 millionparticles, the LPDM took about 30% of the total computation time. From this 30 % theAdams-Bashfort 2nd order integration scheme took about 10% of the time, while the inter-polation schemes required 20% of the time. Unfortunately, there are many parameters thatcan influence the performance of the LPDM. Especially the write statements can take a lotof time and memory. In the simulation described above, there were no write-statements in-cluded, only at the end something needed to be written. In the case of preparing a datasetfor postprocessing, it is likely that the particles will be written every few seconds (LES time).With simulations of the LPDM, files have been created with a size of several Gigabytes, al-though this is in its turn very dependent on the format that is used to store the data and ofcourse, of the amount of data that is written. A line of data contains at least the time, the

39


0.004

0.005

0.006

0.007

0.008

0.009

0.01

4 6 8 10 12

abso

lute

val

ue o

f vo

rtic

ity (

s-1)

time (hours)

|ω2|, with sgs|ω2|, without sgs

|ω2|, center of gridbox|ω2|, edge of gridbox, without bottom

|ω2|, edge ofgridbox, with bottom

Figure 4.16: The absolute vorticity, averaged over the entire domain for simulations with the LPDM.There are 5 different lines plotted. In the first 3 simulations, particles are released at the center ofthe gridboxes. In the following simulation, the particles start at the upper edge of the gridboxes (i.e.point G in figure 3.2). In the last simulation, the particles start at the upper edge of the cells, but thereare also particles at the lower edge of the lowest gridboxes. In the first 2 simulations, the particleswere allowed to move, whereas in the rest of the simulations they were kept at the starting position.The simulations lasted 10 hours, where the particles were inserted after 3 hours. This is a BOMEXcase.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.005 0.01 0.015 0.02 0.025 0.03

z/z i

(-)

absolute value of vorticity (s-1)

AdamsBashfort, sgs onAdamsBashfort, sgs off

Euler, sgs onEuler, sgs off

center, fixed positionedge, fixed position

Figure 4.17: A snapshot of the horizontally averaged, absolute vorticity. The 4 different (althoughnot distinguishable) lines denote two integration schemes, both with and without the sgs-schemeturned on. In the rest of the simulations the particles were kept at the starting position. In the first 5simulations, particles are released at the center of the gridboxes. In the last simulation, the particlesstart at the upper edge of the gridboxes (i.e. point G in figure 3.2). This is a BOMEX case simulation.The snapshot was taken after 2 hours of simulation with the LPDM, where the particles were insertedafter 3 hours. The height is normalized with the height of the inversion, zi = 2000 m.

40


position and the velocities of a particle, but it is possible to write a lot more data, like thehigher order moments of the velocity, the sgs-TKE at the particle position, etcetera.

41


42

Chapter 5

Conserved variable diagrams

In the world of atmospheric scientists, there exists a controversy that has lasted for over 5decades now. The question is whether mixing of a cloud and the surrounding air can bedescribed best by lateral mixing or by vertical mixing, figure (5.1). There is no discussionabout the large part of air entering the cloud from below, since a rising thermal is generallyaccepted as the initiator of the formation of a cloud. With lateral mixing, environmental airis mixed into a cloud, mainly from the sides. While on the opposite, with cloud-top mix-ing, there is only mixing (between cloud and environment) at the bottom and at the top ofa cloud. A cloud will always have properties that are based on the environmetal air origi-nating from below the cloud. The question is now, whether a cloud also has properties ofenvironmental air originating from the top of the cloud, or also from the sides of the cloud.The answer to this question is vital in the modeling of clouds. More information, along withprevious studies on this topic can be found in: e.g. Rogers and Yau [27], Blyth et al. [2], Blyth[1], Jonas [12], Paluch [22] and Squires [30]. In this chapter, first the arguments for lateralmixing will be discussed. After this, the arguments for cloud-top mixing will be examined.With the LPDM, it might be possible to create clear evidence for either one of these theories,at least within the LES framework.

Figure 5.1: The different views on mixing between cloud and environment. The left picture showsthe concept of lateral mixing, whereas on the right the concept of cloud-top mixing is visualized.

43

5.1 Lateral mixing

Figure 5.2: The number of particles entering or leaving a cloud as a function of relative height. Thedata is taken over all clouds in the domain and is sampled every 30 seconds and averaged over aperiod of 4 hours. This is a BOMEX-case simulation.

5.1 Lateral mixing

In order to investigate the mixing properties of a cloud, a simulation was performed, con-taining 1,3 million particles. The particles were released uniformly throughout the domainand were allowed to move freely with the flow. Afterwards it was then possible to backtrackthe trace of the particles, in order to determine whether they had entered the cloud at theside, at the bottom or at the top. The results of this simulation are shown in figure 5.2. Thisfigure shows the number of particles that enters or leaves a cloud, as related to the relativeheight. This relative height is defined as the ratio of the particle height within the cloud,divided by the total cloud height. The pulse of particles entering the cloud from belowcan be clearly distinguished. Furthermore, it can be seen in the figure that there are parti-cles passing the boundaries of a cloud at every height. There is no pulse of particles crossingthe border of a cloud at the top. This could be an indication that lateral mixing is important.

5.2 Cloud-top mixing

An argument in favor of the cloud-top mixing version, is a conserved variable diagram, alsocalled a Paluch diagram (Paluch [22]). In such a diagram, the total water content qt is plottedagainst the liquid water potential temperature, θl. It is shown, that the assumption of linearmixing holds for these conserved variables. Now suppose that there is mixing between twoparcels of air, with different values for qt and θl. Then, because they are conserved variables,the mixture will have values for qt and θl that are directly proportional to the mixing ratio.This is shown for qt in equation 5.1, where 2 parcels of air are mixed (1 and 2) to deliver the

44

5. Conserved variable diagrams

Figure 5.3: The original conserved variables diagram, as presented by Paluch [22]. The Q is the totalliquid water content qt. SFC is the surface at 1.4 km. Cloud bottom CB is at 3.8 km. Cloud top was at9-12 km and the observation level was at 5.2 km.

new parcel of air (3). The α is the mixing ratio.

qt,3 = αqt,1 + (1 − α)qt,2 (5.1)

In 1979, Paluch was the first to do this analysis, by analysing data obtained from aircraftmeasurements. The results of her analysis are shown in figure 5.3. In this case, the wetequivalent potential temperature, θq, is used as a conserved variable instead of θl. θq isgiven as follows:

θq ≈ θl +L

cp

(p

p0

)−(Rd/cp)

qt ≈ θl + 2.5 × 103qt. (5.2)

The reason for this change is that the original conserved variable, θl is strongly correlatedwith ql (see equation 5.3). According to equation 5.4, ql is in turn strongly correlated withqt, so a plot of qt against θl would be biased by this correlation. A derivation of this formulacan be found in appendix B.

θl ≈ θ − L

cp

(p

p0

)−

Rdcp

ql, (5.3)

qt = ql + qv. (5.4)

In figure 5.3 the horizontally averaged environmental curve is plotted as a line. The mea-sured cloud points are also plotted. It turns out that the cloud points are on a line, from the

45

5.3 The choice of conserved variables

2

4

6

8

10

12

14

16

18298 300 302 304 306 308 310 312 314

qt (g

/kg

)

θx (K)

CB

CT

Figure 5.4: A conserved variable diagram for one sampled cloud. Here θx = θl. The line representsthe environmental values, whereas the points are cloudy points. The crosses represent heights ofrespectively 1000 and 1400 m. Also drawn are the positions of cloudbase at 550 m and cloud top at1750 m. The sample is taken after 1 hour of simulation with the LPDM, which has started after 4hours of simulation by the LES.

environmental curve at about 1.5 km to the environmental curve at 8 km. This led Paluch tothe conclusion that the measured cloud points contained a mixture of the properties of airfrom below cloud base and air that originates from a height of 8 km. This result from Paluch(and others) has been used as the main argument for the cloud-top mixing theory. In theremainder of this chapter, in similar figures the y-axis will be reversed, so that if y increases,also the height increases.


In the previous section, it was mentioned that the use of θl would lead to a strongly cor-related result. Unfortunately, it turned out that in this case, the use of the wet equivalentpotential temperature θq leads to another correlation. This can be explained by looking atfigure 5.3. If, from this picture, the environmental curve is taken for a height from 0 to 3.2 km

(as is the height of the BOMEX-case), then the environmental curve turns out to be a straightline. The mentioned correlations can be seen in figures 5.4 to 5.9. Here six conserved vari-able diagrams are shown, with the same data, but with different values for θx. This variableis defined as follows:

θx ≡ θl + cx103qt. (5.5)

The prefactor is dependent on the unit of qt; it is 103 because qt is given in [g/kg]. The cx isa constant that has to be chosen so that the figures become less correlated. Note that cx = 0gives θx = θl and cx = 2.5 gives θx = θq. The cx constant was varied from 0 to 2.5. In figure5.6 the curvature is maximal, so the value cx = 0.7 is used in the remainder of the research.Despite the fact that there is no physical background for the use of this variable, θx can beused, because it is a linear combination of two conserved variables and is therefore also aconserved variable.

46


6

8

10

12

14

16

18308 308.5 309 309.5 310

qt (g

/kg

)

θx (K)

CB

CT

Figure 5.5: A different form of the same conserved variable diagram. Here θx = θl + 0.6 × 103qt.The line represents the environmental values, whereas the points are cloudy points. The crossesrepresent heights of respectively 1000 and 1400 m. Also drawn are the positions of cloudbase at 550m and cloud top at 1750 m. Output is given after 1 hour of simulation with the LPDM, which hasstarted after 4 hours of simulation by the LES.

6

8

10

12

14

16

18309.6 309.8 310 310.2 310.4 310.6 310.8 311

qt (g

/kg

)

θx (K)

CB

CT


47


2

4

6

8

10

12

14

16

18311 312 313

qt (g

/kg

)

θx (K)

CB

CT


2

4

6

8

10

12

14

16

18312 313 314 315 316 317

qt (g

/kg

)

θx (K)

CB

CT

Figure 5.8: A different form of the same conserved variable diagram. Here θx = θl + 103qt. The linerepresents the environmental values, whereas the points are cloudy points. The crosses representheights of respectively 1000 and 1400 m. Also drawn are the positions of cloudbase at 550 m andcloud top at 1750 m. Output is given after 1 hour of simulation with the LPDM, which has startedafter 4 hours of simulation by the LES.

48


2

4

6

8

10

12

14

16

18315 320 325 330 335 340 345

qt (g

/kg

)

θx (K)

CB

CT

Figure 5.9: A different form of the same conserved variable diagram. Here θx = θl +2.5×103qt ≈ θq .The line represents the environmental values, whereas the points are cloudy points. The crossesrepresent heights of respectively 1000 and 1400 m. Also drawn are the positions of cloudbase at 550m and cloud top at 1750 m. Output is given after 1 hour of simulation with the LPDM, which hasstarted after 4 hours of simulation by the LES.

5.4 The setup of the case

In order to answer the question, whether mixing of a cloud and the surrounding air can bedescribed best by lateral mixing or by vertical mixing, a simulation was performed, con-taining over 1 million particles (128×128×80). The particles were released after 4 hours ofsimulation by the LES. After 1 hour, the biggest cloud was selected and all particles thatwere at that time inside the cloud, were selected. A projection of the selected cloud is givenin figure 5.10. There were 1731 particles selected. After the selection procedure, the sim-ulation went on for another hour. Since the particle data was written every 30 seconds, itwas possible to track the positions of the particles offline. Because all calculated variableswere stored with the position of the particles, it was also possible to visualize the particlesin a conserved variable diagram. The sample heights of 800, 1000 and 1400 m are shown infigure 5.10. For this cloud, cloudbase is at 550 m and cloud top at 1750 m. In order to ensurea sufficient number of particles, the particles were selected within a range of 40 m in height,centered around the observation heights of 800 m and 1000 m. A range of 20 m in height wasused for the observation level of 1400 m. The cloud sample at 800 m contains 45 particles,the cloud sample at 1000 m contains 34 particles and the cloud sample at 1400 m contains 66particles. The lowest two cloud samples contain a sample of a rising cloud, whereas at 1400m the cloud is close to the top and the sample contains both rising air and descending air.The selected particles can be visualized in a Paluch diagram, at different timesteps.

5.5 Results of the case

The analysis of the conserved variable diagram, as introduced by Paluch, often leads to con-clusions that are pointing towards cloud-top mixing as the dominant mixing process. In thissection it will be shown, that it is possible to give an argument towards lateral mixing as thedominant mixing process in cumulus clouds, by using a Paluch diagram. In figures 5.11,

49

5.5 Results of the case

400

600

800

1000

1200

1400

1600

1800

800 1000 1200 1400 1600 1800 2000 2200

heig

ht (

m)

x (m)

selected cloud points

400

600

800

1000

1200

1400

1600

1800

1000 1200 1400 1600 1800 2000 2200

heig

ht (

m)

y (m)

selected cloud points

Figure 5.10: The projection of the selected cloud. The lines represent the sample heights. Selectiontime was after 1 hour of simulation with the LPDM, which has started after 4 hours of simulation bythe LES.

5.12, 5.13 and 5.14 the history of the particles is shown in a Paluch diagram. This is done fora group of particles that is selected to be in the same cloud, at the same height, at the sametime. If the conclusion of Paluch were true, then the particles should originate either fromthe bottom of the cloud, or from the top of the cloud. This would mean that the particlesshould spread in the Paluch diagram on a line towards cloud top or towards cloud bottom.For the sample heights of 800 m and 1000 m it can be seen that the particles do not do so.Instead they seem to originate from various positions approximately on the environmentalcurve, with a somewhat stronger correlation towards cloud bottom. This means that theparticles were mixing laterally with environmental air, while there also were some particlesoriginating from cloud bottom. The latter was not under discussion, since a rising thermalfrom the bottom is generally believed to be the cause of the existence of a cloud. However,the first conclusion is exactly the proof that is needed to make the statement, that lateralmixing is a better way to describe the mixing of environmental air into a cloud, than cloud-top mixing is.

However, for the sample height of 1400 m the results are pointing towards a different con-clusion. There is an amount of particles originating from above observation height. Thismeans that higher in the cloud, there is some influence from the top. This can be explainedby the fact that a cloud contains more turbulence near the top and therefore some influencefrom cloud top can be expected higher in the cloud. In figure 5.15 there are two lines plottedthat show the two influences. For cloudy particles that are beneath 1000 m, the line that canbe drawn through these points is clearly a straight line towards a height of approximately800 m. The other line states that cloudy points higher than 1000 m are mixed with environ-mental air originating from a height of about 1650 m. These lines become even more visibleif the values for the cloudy points are averaged over the height. This is done in figure 5.16.

The conclusions as described above is even more supported by the figures 5.17, 5.18 and5.19. In these figures the previous heights of the same particles can be seen. In the case ofcloud-top mixing, the particles should originate from heights that are well above observa-tion height, or well below observation height. It can be seen from the figures, that for thesample heights of 800 m and 1000 m, the particles originate from all heights, with the ma-jority originating from heights just below the observation height. For the sample height of1400 m, there are also particles coming from above observation height. This also gives good

50


evidence for the lateral mixing concept to be the most appropriate in the bulk of the selectedcumulus cloud. Only near the top there is some influence of a vertical mixing process.

5.6 Applicability of the results

Some remarks can be made of the results mentioned above. It is already mentioned that adifference in cloud height can affect the results in a conserved variable diagram. It is alsopossible that different types of clouds will have different mixing characteristics. Anothertopic of importance is the lifecycle of a cloud. A young growing cloud consists of a stronglybuoyant updraft, with a lot of liquid water. This in contrast to a mature cloud, which haslittle liquid water left and has almost zero buoyancy. It can be expected that the mixingproperties of such a young cloud, are different from the mixing properties of a large, maturecloud. A complication is also that, in growing clouds, it is hard to distinguish between lateralmixing and mixing at the increasing top. More research is needed in order to investigate themixing properties of the different types and ages of clouds.

51


6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t - 5 minutes

CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)θx (K)


CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t

CB

CT

Figure 5.11: The history of the cloudy points at the sample height of 800 meter. Here θx = θl+0.7×qt.The sample time is at 1 hour after the restart of the simulation. The restart took place after 4 hoursof simulation. The line represents the environmental values, where a cross is drawn at the sampleheight. The cloudy points are visualized by points if they are a cloud and by a plus sign if they arenot. Also visible are: cloud-base at 550 m and cloud-top at 1750 m.

52


6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)θx (K)


CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t

CB

CT


53


6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)θx (K)


CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)


CB

CT6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t

CB

CT


54


6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t - 1 hour

CB

CT 800 m6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)θx (K)

time = t

CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t - 1 hour

CB

CT 1000 m6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t

CB

CT

6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t - 1 hour

CB

CT 1400 m6

8

10

12

14

16

18309.5 310 310.5 311

qt (g

/kg

)

θx (K)

time = t

CB

CT

Figure 5.14: The history of the cloudy points at the different sample heights. Here θx = θl + 0.7 × qt.The sample time is at 1 hour after the restart of the simulation. The restart took place after 4 hoursof simulation. The line represents the environmental values, where a cross is drawn at the sampleheight. The cloudy points are visualized by points if they are a cloud and by a plus sign if they arenot. Also visible are: cloud-base at 550 m and cloud-top at 1750 m.

55


6

8

10

12

14

16

18309.6 309.8 310 310.2 310.4 310.6 310.8 311

qt (g

/kg)

θx (K)

CB

CT

Figure 5.15: Two linear fits through the total of 1731 selected cloudy points, at 1 hour after the restartof the simulation. The two lines cross at a height of 980 m. Here θx = θl + 0.7 × qt. The restart tookplace after 4 hours of simulation. The line represents the environmental values, where the crossesare drawn every 100 m, starting with 700 m. The cloudy points are visualized by points, also visibleare: cloud-base at 550 m and cloud-top at 1750 m.

6

8

10

12

14

16

18309.6 309.8 310 310.2 310.4 310.6 310.8 311

qt (g

/kg)

θx (K)

CB

CT

Figure 5.16: Two linear fits through the horizontally averaged values of the cloudy points, at 1 hourafter the restart of the simulation. The two lines cross at a height of 980 m. Here θx = θl +0.7×qt. Therestart took place after 4 hours of simulation. The curved line represents the environmental values,where the crosses are drawn every 100 m, starting with 700 m. The cloudy points are visualized bypoints, also visible are: cloud-base at 550 m and cloud-top at 1750 m.

56


0

500

1000

1500

2000

-10 -8 -6 -4 -2 0 2 4 6 8 10

heig

ht (

m)

time (minutes)

-- CB

-- OB

-- CT

cloudenvironment

Figure 5.17: The time dependent heights of particles that are present in a cloud at an observationheight (OB) of 800 m, at sample time. The sample time is at 1 hour after the restart of the simulation.The restart took place after 4 hours of simulation. Cloud base (CB) is indicated at 550 m, cloud top(CT) is shown at 1750 m.

0

500

1000

1500

2000

-10 -8 -6 -4 -2 0 2 4 6 8 10

heig

ht (

m)

time (minutes)

-- CB

-- OB

-- CT

cloudenvironment

Figure 5.18: The time dependent heights of particles that are present in a cloud at an observationheight (OB) of 1000 m. The sample time is at 1 hour after the restart of the simulation. The restarttook place after 4 hours of simulation. Cloud base (CB) is indicated at 550 m, cloud top (CT) is shownat 1750 m.

57


0

500

1000

1500

2000

-10 -8 -6 -4 -2 0 2 4 6 8 10

heig

ht (

m)

time (minutes)

-- CB

-- OB

-- CT

cloudenvironment

Figure 5.19: The time dependent heights of particles that are present in a cloud at an observationheight (OB) of 1400 m. The sample time is at 1 hour after the restart of the simulation. The restarttook place after 4 hours of simulation. Cloud base (CB) is indicated at 550 m, cloud top (CT) is shownat 1750 m.

58

Chapter 6

Conclusions and recommendations

This chapter will present some conclusions from this thesis, along with some remarks andrecommendations. First the LPDM will be discussed, as it is implemented and running.After this, some attention will be given to the subgrid scheme and finally some results thatare obtained with the LPDM will be discussed.

6.1 The LPDM

This section contains the conclusions that are directly related to the implementation andtesting of the LPDM.

6.1.1 Implementation of the LPDM

The first conclusion is that the LPDM is implemented and working without problems. Itis now possible to follow an amount of particles through time, as they travel within thedomain. The LPDM has been proven to work with a total of over 10 million particles. TheLPDM runs parallel to the LES and uses the resolved velocities from the LES. The particlescheme has to interpolate these velocities for each particle and then the new positions ofthe particles are calculated using the integration scheme and the subgrid scheme. At everytimestep the boundary conditions are checked. These topics will be discussed in more detailin the following paragraphs.

The integration scheme In the production runs of the LPDM, usually the Adams-Bashfort2nd order scheme was used. Although it was expected in advance that the 2nd order schemewould give better results than the 1st order scheme, the difference turned out to be rathersmall. The interpolation errors were shown to be dominating the results. Furthermore, sincethe velocities that are the input of the integration scheme, are filtered in the LES, there areno rapid changes in the velocity possible. Therefore the 2nd order Adams-Bashfort schemewill be good enough for current use, although it is always possible to implement a 4th orderscheme, if after improvement of the interpolation scheme, it turns out to be necessary.

The spatial interpolation scheme In the LPDM the 1st order tri-linear interpolation schemeis used. Also because the velocities are filtered in the LES, it was not expected that there arelarge spatial variations in the domain. However, it turned out that in calculating derivatives,it mattered a lot whether they were calculated in the center of a gridbox, or at an edge of the

59

6.2 The LPDM in practice

gridbox. The conclusion is that the simple interpolation scheme is good enough for velocitycalculations, but that it leads to large errors in calculations of derivatives.

The subgrid scheme It was found that an extra subgrid scheme was needed for the par-ticles, because the input to the particle subroutine consists of resolved velocities, that areunable to fully describe the particle motions. It was especially needed to avoid a build upof particles at the bottom of the domain. A stochastic subgrid scheme that has been adaptedto work inside an LES was implemented.

The boundary handling scheme The LPDM has the same boundary conditions as the LES,except that the particle scheme has perfect reflection implemented at the top and bottom ofthe domain. The latter requires some explanation, since massless particles normally do nothit the walls. The subgrid scheme is totally responsible for any particles running into thewalls. It can be concluded that the boundary conditions do not give any problems in theLPDM

6.2 The LPDM in practice

6.2.1 Validation of the LPDM scheme

In order to validate the particle scheme and to make sure that it was implemented correctly,some tests were performed. After the comparison of several quantities that were calculatedboth by the LES and by the LPDM, it can be concluded that the LPDM is working as itshould. Particles that are uniformly spread in the domain, should remain uniformly dis-tributed. Without the subgrid scheme there was a number particles remaining at the bottomof the domain. This surplus of particles was continuously increasing to over 40% after just 2hours of simulation. The use of the subgrid scheme has been shown to keep the surplus ofparticles at the bottom of the domain at approximately 5%. This is good enough for currentuse.

6.2.2 Computation time of the LPDM

With a simulation that contained over 10 million particles, the LPDM took about 30% of thetotal computation time. This is acceptable for current usage. Unfortunately, there are manyparameters that can influence the performance of the LPDM. Especially the write statementstake a lot of time and memory. These statements are very dependent on the format that isused to store the data and of course of the amount of data that is written. A line of datacontains at least the time, the position and the velocities of a particle, but it is possible towrite a lot more data, like the higher order moments of the velocity, the TKE at the particleposition, etcetera.

6.2.3 The timestep of the LPDM

In the LPDM the timestep was set to be equal to the timestep of the LES. This means that inthe test cases, a timestep of 2 seconds was used and in the production runs, a timestep of 1second was used. Since the requirement for the timestep of the LES is more or less equal tothe requirement of the LPDM, it seems reasonable to put the timestep of the LPDM at the

60

6. Conclusions and recommendations

same value as the timestep of the LES. However the LPDM can handle almost every valueof the timestep, as long as it is either a multiple of the timestep of the LES, or vice versa.

6.2.4 The κ-scheme and the subgrid scheme

Unfortunately, the implemented sgs-scheme becomes unstable if a κ-advection scheme isused. This is due to the dissipative properties of the κ-scheme, which makes that the fs

parameter goes to zero, where it should not. This fs parameter is calculated by dividingthe subgrid variances by the total variances. This problem is solved by manually setting thefs parameter to 1. The LPDM is thus positioned in the middle of the models as posed byWeil et al. [37] and Thomson [34]. However, the change of this parameter could have someinfluence on statistics.

6.3 Lateral or vertical mixing

In a conserved variable diagram, two conserved variables (θq and qt) are plotted againsteach other. Such a diagram seems to be a major argument for top-bottom mixing as the bestway to describe mixing of cloudy air with its environment. Recent research has come upwith evidence towards lateral mixing. This conclusion is examined by looking at the timedependent conserved variable diagram, by means of an experiment with the LPDM.

6.3.1 Conserved variable diagram

It was possible to construct a conserved variable diagram, in the same way as Paluch [22].The only problem was that the cumulus clouds that are investigated here, are much smallerthan the clouds investigated by Paluch. This results in a diagram that contains a strongcorrelation between the two variables θq and qt. This problem is solved by using a differentconserved variable, θx ≡ θl + 0.7 × 103qt, which has no physical background, but, since it iscomposed of a linear combination of two conserved variables, can be used legitimately.

6.3.2 Setup and results from the BOMEX-case

Simulations were performed on a 256× 256× 160 grid, containing over 1,3 million particles.With the generated data, the particles could be followed back in time to see where they camefrom. If the conclusion of top-bottom mixing being the dominant mixing process were true,then the particles should originate either from the bottom of the cloud, or from the top ofthe cloud. It follows from the results, that the particles do not do so. Instead, for the sampleheights of 800 m and 1000 m, they are found to originate from various positions approx-imately at the observation height, on the environmental curve, with a somewhat strongercorrelation towards cloudbottom. This means that for these sample heights, the particleswere mixing laterally with environmental air, while also some particles were originatingfrom cloud bottom. For the sample height of 1400 m, there seem to be two processes actingon the cloud, since there are also particles originating from above observation level. Theseresults clearly show that also in the form of a conserved variable diagram, lateral mixingcan be seen as the dominant mixing process for the bulk of the cloud. Only near the top ofthe cloud, there seems to be an influence from vertical mixing from the top of the cloud.

61

6.4 Future research

6.3.3 Comments on the results

A difference in cloud height can affect the results in a conserved variable diagram. It is alsopossible that different types of clouds will have different mixing characteristics. Anothertopic of importance is the lifecycle of a cloud. In a young growing cloud, it can be expectedthat the mixing properties are different from the mixing properties of a large, mature cloud.A complication is also that in a growing cloud, it is hard to distinguish between lateralmixing and mixing at the increasing top. More research is needed in order to investigate themixing properties of the different types and ages of clouds.

6.4 Future research

In the future, it might be necessary to make improvements based on the different parts ofthe LPDM. A more sophisticated interpolation scheme would be cubic splines. For the inte-gration method, the 4th order Runge-Kutta scheme might be implemented. In the subgridscheme there is the fs parameter that is manually set to 1. Some time has to be spent in orderto see if there is a better solution to make the subgrid scheme stable in combination with theκ-scheme. It is believed, that with the addition of a Monin-Obukhov scheme for the bottomgrid cells, the surplus of particles at the bottom of the of the domain can be further reduced.A Monin-Obukhov scheme would include an approximately logarithmic decay of velocitiestowards the bottom of the domain, instead of the present linear decay. Also the LPDM cur-rently works only with an equidistant grid. In the future adaptions might be made to makethe LPDM compatible with non-equidistant grids. Future research is also needed in orderto determine whether it is worthwhile to use a larger timestep for the LPDM. It is not surehow much the combined LPDM-LES would run faster and what would be the penalty inaccuracy for choosing a larger timestep for the LPDM. With regard to the Paluch diagrams,the next step would be to look at more clouds while using even more particles, in order toobtain better statistics. Furthermore, the output from the LPDM might be visualized in avirtual environment. Research towards this is already being performed and promises to beof great utility.

62

Appendix A

Derivation of potential temperature.

The following equations are used in the derivation of the potential temperature:

pv = RdT, (A.1)

dh = vdp, (A.2)

dh = cpdT. (A.3)

These equations are the ideal gas law and two equations for a change in enthalpy, h. In theseequations p is the pressure, Rd is the gas constant for dry air, T is the temperature, v is thespecific volume of air and cp is the heat capacity of air. Combining all three equations gives:

cpdT =RT

pdp (A.4)

Rearranging delivers:

dT

T=

R

cp

dp

p(A.5)

Integration of this equation gives:

T

T0=

(p

p0

) Rcp

(A.6)

Now the potential temperature is defined as the temperature a parcel of air would have if itwas brought adiabatically to the reference pressure p0:

θ ≡ T0 = T

(p

p0

)−

Rcp

(A.7)

63

Appendix B

Derivation of the wet equivalentpotential temperature

The wet equivalent potential temperature θq, as used in equation 5.2, can be derived fromthe definition of θq given by Paluch [22]:

θq = T

(p

p0

)−(Rd/Cp,a)γ

exp[qvLγ

Cp,aT], (B.1)

γ =1

1 +Cp,w

Cp,aql

.

The variables are as defined in chapter 2. With Cp,w ≈ 4Cp,a and ql ≈ 10−2, then γ ≈ 1. If theapproximation of γ is used and with the application of the expansion for ex = 1+x+O(x2),then the equation can be approximated by:

θq = T

(p

p0

)−(Rd/Cp,a)

[1 +qvL

Cp,aT]. (B.2)

This equals:

θq = T

(p

p0

)−(Rd/Cp,a)

+L

Cp,a

(p

p0

)−(Rd/Cp,a)

qv. (B.3)

With qv = qt − ql, this becomes:

θq = T

(p

p0

)−(Rd/Cp,a)

+L

Cp,a

(p

p0

)−(Rd/Cp,a)

qt −L

Cp,a

(p

p0

)−(Rd/Cp,a)

ql. (B.4)

Now the first and the last term together form θl (equation 2.6):

θq = θl +L

Cp,a

(p

p0

)−(Rd/Cp,a)

qt. (B.5)

The following values are used:

• L = 2.5 × 106Jkg−1

• Cp,a = 1004Jkg−1K−1

65

• p ≈ 9 × 104Pa

• p0 = 105Pa

• Rd = 287Jkg−1

With these values the final equation can be given:

θq ≈ θl + 2.5 × 103qt. (B.6)

66

Appendix C

Details of the time consumption ofseveral parts of the LPDM

This appendix contains information about the performace of the LPDM. Because of the enor-mous amount of information, only the results from 1 processor are given. The usertime rou-tine gives lots of details of internal processes, so for clearity, only the data that took morethan 1 % of the computation time is presented.

-------------------------------------------------------------------------SpeedShop profile listing generated Wed Dec 6 12:23:47 2006

Summary of statistical callstack sampling data (usertime)--1250762: Total Samples

0: Samples with incomplete traceback37522.860: Accumulated Time (secs.)

30.0: Sample interval (msecs.)-------------------------------------------------------------------------Function list, in descending order by exclusive time-------------------------------------------------------------------------excl.% cum.% incl.% procedure (dso: file, line)

9.5% 9.5% 29.0% particles (lesv2_0.209: particles.f90, 263)9.4% 18.8% 10.2% PARTCOMMUNICATE (lesv2_0.209: modparticles.f90, 179)6.9% 25.7% 7.9% thermo (lesv2_0.209: thermo.f90, 3)6.9% 32.6% 6.9% sources (lesv2_0.209: sources.f90, 2)6.3% 38.9% 6.3% __sginap (libc.so.1: flush.c, 321; compiled in sginap.s)5.5% 44.4% 5.5% memcpy (libc.so.1: stat.c, 32; compiled in bcopy.s)5.0% 49.5% 5.0% calthv (lesv2_0.209: calthv.f90, 1)4.0% 53.4% 6.1% all_all_j (lesv2_0.209: all_all_j.f90, 3)2.7% 56.1% 2.7% tstep (lesv2_0.209: tstep.f90, 2)2.6% 58.8% 3.1% ADAMSBASH2 (lesv2_0.209: modintscheme.f90, 93)2.6% 61.3% 2.6% diffw (lesv2_0.209: diffw.f90, 3)2.5% 63.8% 5.1% MPI_SGI_progress (libmpi.so: progress.c, 78)2.5% 66.3% 4.5% fillps (lesv2_0.209: fillps.f90, 2)2.2% 68.6% 2.2% VELOCITY_URES (lesv2_0.209: modpartfunc.f90, 13)2.2% 70.8% 2.4% MPI_SGI_shared_progress (libmpi.so: shared.c, 1577)2.1% 72.8% 2.1% VELOCITY_VRES (lesv2_0.209: modpartfunc.f90, 88)2.0% 74.8% 2.0% VELOCITY_WRES (lesv2_0.209: modpartfunc.f90, 163)1.8% 76.6% 1.8% advecc (lesv2_0.209: advecc.f90, 2)1.6% 78.2% 4.3% excjs (lesv2_0.209: excjs.f90, 3)1.6% 79.9% 1.6% diffc (lesv2_0.209: diffc.f90, 3)1.4% 81.3% 1.4% diffv (lesv2_0.209: diffv.f90, 3)1.4% 82.7% 1.4% diffu (lesv2_0.209: diffu.f90, 3)1.2% 83.9% 8.4% solmpj (lesv2_0.209: solmpj.f90, 2)1.1% 85.0% 3.6% excj (lesv2_0.209: excj.f90, 2)1.0% 86.0% 1.0% __exp (libm.so: exp.c, 103)1.0% 87.0% 94.5% les (lesv2_0.209: program.f90, 2)

67

Bibliography

[1] Alan M. Blyth. Entrainment in cumulus clouds. Journal of applied meteorology, 32:626–641, 1993.

[2] Alan M. Blyth, William A. Cooper, and Jorgen B. Jensen. A study of the course ofentrained air in montana cumuli. Journal of the atmospheric science, 45:3944–3964, 1988.

[3] M.S. Borgas and B.L. Sawford. Stochastic equations with multifractal random incre-ments for modeling turbulent dispersion. Physics of Fluids, 6:618–633, 1994.

[4] David Brickman and P.C. Smith. Lagrangian stochastic modelling in coastal oceanog-raphy. Journal of atmospheric and oceanic technology, 19:83–99, 2002.

[5] D.L. Darmofal and R. Haimes. An analysis of 3-d particle path integration algorithms.In AIAA CFD Meeting, San Diego, California, 1995.

[6] S.K. Das and P.A. Durbin. A lagrangian stochastic model for dispersion in stratifiedturbulence. Physics of fluids, 17:025109 (1–10), 2005.

[7] Shuming Du, John D. Wilson, and Eugene Yee. On the moments approximation methodfor constructing a lagrangian stochastic model. Boundary-layer meteorology, 40:273–292,1994.

[8] Pasquale Franzese, Ashok K. Luhar, and Micheal S. Borgas. An efficient lagrangianstochastic model of vertical dispersion in the convective boundary layer. Atmosphericenvironment, 33:2337–2345, 1999.

[9] J.R. Garratt. The atmospheric boundary layer. Cambridge University Press, 1994.

[10] S.G. Gopalakrishnan and R. Avissar. A les study of the impacts of land surface het-erogeneity on dispersion in the convective boundary layer. Journal of the atmosphericsciences, 57:352–371, 1999.

[11] W. Hundsdorfer, B. Koren, M. van Loon, and J.G. Verwer. A positive finite-differenceadvection scheme. Journal of computational physics, 117:35–46, 1995.

[12] P.R. Jonas. Observations of cumulus cloud entrainment. Atmospheric research, 25:105–127, 1989.

[13] Ashok k. Luhar and Rex E. Britter. A random walk model for dispersion in inhomo-geneous turbulence in a convective boundary layer. Atmospheric environment, 23:1911–1924, 1989.

69

BIBLIOGRAPHY

[14] David N Kenwright and David A. Lane. Interactive time-depandent particle tracingusing tetrahedral decomposition. In IEEE TVCG, volume 2(2), page 120?129, 1996.

[15] S. Kim, C.H. Moeng, J.C. Weil, and M.C. Barth. Lagrangian particle dispersion mod-eling of the fumigation process using large-eddy simulation. Journal of the AtmosphericSciences, 62:1932–1946, 2005.

[16] N. Kljun, M.W. Rotach, and H.P. Schmid. A three dimensional backward lagrangianfootprint model for a wide range of boundary-layer stratifications. Boundary-Layer Me-teorology, 103:205–226, 2002.

[17] K. Kontomaris, T.J. Hanratty, and J.B. McLaughlin. An algorithm for tracking fluidparticles in a spectral simulation of turbulent channel flow. journal of computationalphysics, 103:231–242, 1992.

[18] Igor Kos, Danijel Belusic, Amela Jericevic, Kristian Horvath, Darko Koracin, andMaja Telisman Prtenjak. Education and research: Initial development of the atmmo-spheric lagrangian particle stochastic (alps) dispersion model. Geofyzika, 21:37–52, 2004.

[19] R.G. Lamb. A numerical simulation of dispersion from an elevated point source in theconvective planetary boundary layer. Atmospheric environment, 12:1297–1304, 1978.

[20] D. Lane. Scientific visualization of large scale unsteady fluid flow, 1996.

[21] P.J. Mason. Large-eddy simulations of dispersion in convective boundary layers withwind shear. Atmospheric environment, 26A:1561–1571, 1992.

[22] I.R. Paluch. The entrainment mechanism in colorado cumuli. Journal of the atmosphericsciences, 36:2467–2478, 1979.

[23] S.B. Pope. Turbulent flows. Cambridge University Press, 2000.

[24] K. Shankar Rao. Lagrangian stochastic moddeling of dispersion in the stable boundarylayer. Boundary-Layer Meteorology, 90:541–549, 1999.

[25] A.M. Reynolds and G. Lo Iacono. On the simulation of particle trajectories in turbulentflows. physics of fluids, 16:43534358, 2004.

[26] U. Rizza, C. Mangia, J.C. Carvalho, and D. Anfossi. Estimation of the lagrangian veloc-ity structure function constant c0 by large-eddy simulation. Boundary Layer Meteorology,120:25–37, 2006.

[27] R. R. Rogers and M. K. Yau. A short course in cloud physics. Butterworth-Heinemann,1989.

[28] B.L. Sawford and P.K. Yeung. Eulerian acceleration statistics as a discriminator betweenlagrangian stochastic models in uniform shear flow. Physics of fluids, 12:2033–2045, 2000.

[29] A. Pier Siebesma, Christopher S. Bretherton, Andrew Brown, Andreas Chlond, JoanCuxart, Peter G. Duynkerke, Hongli Jiang, Marat Khairoutdinov, David Lewellen,Chin-Hoh Moeng, Enrique Sanchez, Bjorn Stevens, and David E. Stevens. A large eddysimulation intercomparision study of shallow cumulus convection. Journal of the Atmo-spheric Sciences, 60:1201–1219, 2003.

70

BIBLIOGRAPHY

[30] P. Squires. Penetrative downdraughts in cumuli. Tellus, 10:381–389, 1958.

[31] A. Stohl, M. Hittenberger, and G. Wotawa. Validation of the lagrangian particle disper-sion model flexpart against large scale tracer experiment data. Atmospheric Environment,32:4245–4264, 1998.

[32] Andreas Stohl and David J. Thomson. A density correction for lagrangian particledispersion models. Boundary-Layer Meteorology, 90:155–167, 1999.

[33] R.B. Stull. An introduction in Boundary Layer Meteorology. Kluwer Academic Publishing,1988.

[34] D.J. Thomson. Criteria for the selection of stochastic models of particle trajectories inturbulent flows. Journal of Fluid Mechanics, 180:529–556, 1987.

[35] Marek Uliasz and Zbigniew Sorbjan. lagrangian statistics in atmospheric boundarylayer derived from les, 1999.

[36] Jon S. Warland and George W. Thurtell. A lagrangian solution to the relationship be-tween a distributed source and concentration profile. Boundary-Layer Meteorology, 96:453–471, 2000.

[37] J.C. Weil, P.P. Sullivan, and C.H. Moeng. The use of large-eddy simulations in la-grangian particle dispersion models. Journal of the atmospheric sciences, 61:2877–2887,2004.

[38] W.E. Willis and J.W. Deardorff. A laboratory study of dispersion from an elevatedsource within a modeled convective planetary boundary layer. Atmospheric Environ-ment, 12:1305–1311, 1977.

[39] John D. Wilson and Brian L. Sawford. Review of lagrangian stochastic models for tra-jectories in the turbulent atmosphere. Boundary-layer meteorology, 78:191–210, 1996.

[40] P.K. Yeung and S.B. Pope. An algorithm for tracking fluid particles in numerical simu-lations of homogeneous turbulence. Journal of computational physics, 79:373–416, 1988.

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